Chapter B, Problem 54E

To determine

To calculate: To sketch the region bounded by the curves

Answer to Problem 54E

The region bounded by the curves is sketched

Explanation of Solution

Given information: Equation of curves are $y=4-x^2$ and $x-2y=2$

Calculation:

Equation of curves are given as follows:

  $y=4-x^2$ --------(1)

  $x-2y=2$ -------(2)

Substituting the value of $y=4-x^2$ in equation (2)

  $\begin{array}{l}x-2y=2\\ x-2\left(4-x^2\right)=2\\ x-8+2x^2=2\\ 2x^2+x-10=0\\ 2x^2+5x-4x-10=0\\ x\left(2x+5\right)-2\left(2x+10\right)=0\\ \left(2x+5\right)\left(x-2\right)=0\\ x=2,-\frac{5}{2}\end{array}$

Now, substituting the values of $x=2$ in equation (1)

  $\begin{array}{l}x-2y=2\\ 2-2y=2\\ -2y=0\\ y=0\end{array}$

Thus, one point of intersection is $\left(2,0\right)$

Again, substituting the values of $x=-\frac{5}{2}$ in equation (1)

  $\begin{array}{l}x-2y=2\\ -\frac{5}{2}-2y=2\\ -2y=2+\frac{5}{2}\\ -2y=\frac{9}{2}\\ y=-\frac{9}{4}\end{array}$

Thus, second point of intersection is $\left(-\frac{5}{4},-\frac{9}{4}\right)$

Hence the area between the curves is displayed as follows:

  

Conclusion:

Hence, region bounded by the curves is sketched